The Korean expression 어찌나 + A/V + 는지 is used to emphasize how extremely a certain quality or attribute is manifested.
This structure can be translated to 'it's so A/V that...'.
It is commonly used to highlight extreme feelings, actions, or conditions.
Examples
Last night after volleyball, I was so tired that I went home and immediately fell asleep.
I was so fed up with my friend's complaints that I asked him to leave.
That guy calls me so often that it annoys me to death.
Those two resemble each other so much that everyone thinks they're siblings.
Usain Bolt is so fast that no one else can match him.
In the summer, the weather was so hot that I had to shower three times a day.
I was so embarrassed that I wanted to die!
Conclusion
The 어찌나 + A/V + 는지 structure allows you to put extra emphasis on the extremeness of a situation or condition.
It is a useful tool for expressing strong emotions, actions, or qualities.